Why it sounds like that.

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

Two notes and a ratio, which is the whole of consonance

Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation.

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.

Three notes at once, and why these three

A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.

Rhythm is a circle, and the bar line is a choice

Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting.

A string does everything at once

A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact.

The quietest thing audible, and why the volume knob is a tone control

A decibel is a fact about air. A phon is a fact about a listener, and the two do not line up — the map between them bends with frequency, and it bends differently at every level. One consequence is that turning a piece of music down does not turn all of it down equally, and the amount by which it does not is a number.

A tube that skips every other partial

Stop one end of a cylinder and half its modes vanish. That single fact about where the pressure has to be decides that a clarinet sounds hollow, that it plays an octave below its length suggests, and that it must cover nineteen semitones with fingers before it can overblow — while every other woodwind covers twelve.

A piece is mostly itself again

Take a piece of music, encode each bar as the notes sounding in it, and compare every bar with every other bar. The picture that comes out has blocks and stripes in it, and those blocks and stripes are the form — arrived at by arithmetic that has never heard of an exposition, a chorus or a refrain.

Progression

1. 1 A progression is a path, and the map can be drawn
2. 2 The progression that never comes home
3. 3 How often the chord changes
4. 4 Which notes are the chord
5. +30 more

Beating

1. 1 Beats are arithmetic that anybody can hear
2. 2 A tuner counts beats, and that is the whole method
3. 3 The other wolf
4. 4 Three strings, and the note that comes back
5. +23 more

Key-relations

1. 1 Keys are neighbours, and the map is computed
2. 2 The key plan is the form
3. 3 Three ways to measure how far a key is
4. 4 How much evidence a modulation needs
5. +23 more

Harmonic series

1. 1 A string does everything at once
2. 2 The piano is tuned wrong on purpose
3. 3 The series is not a chord
4. 4 The series has three tops
5. +22 more

Open strings

1. 1 The tuning a string quartet cannot change
2. 2 The cello cannot hear its own tempering
3. 3 An open string pulls the quartet flat
4. 4 A guitar tuned by harmonics hides a comma
5. +22 more

Beyond twelve

1. 1 A scale is not a set of pitches
2. 2 A scale without an octave, and the spectrum that asks for it
3. 3 The third the model has no opinion about
4. 4 Three answers to how finely a pitch can be heard
5. +20 more

Small whole numbers

Consonance is frequency ratios of small integers, and it has been known since Pythagoras. Almost everything interesting follows from those integers refusing to line up.

How many of the twelve

A second family of small integers, and they are not ratios. Seven notes chosen from twelve, three of those seven stacked, six semitones that split the octave evenly, three beats laid against two. These arguments never mention a frequency: the object is a set of positions on a circle, and the answers are counted rather than measured.

The comma that will not close

Twelve fifths do not equal seven octaves. Every tuning system in history is a different decision about where to hide the difference.

The ear is not a microphone

Loudness is not amplitude, pitch is not frequency, and a note whose fundamental is missing still has one. Perception is a model, not a measurement.

What the listener supplies

A downbeat, a key, a category, a preference — none of them is in the sound, and every one of them can be measured. These essays are about the part of music that is contributed by the person listening, which is also the part that varies between listeners and between traditions.

The limits of the listener

The quietest audible tone, the smallest audible difference, the fastest series that can still be a beat, the delay at which one sound becomes two. Each is a measured boundary with a published source and a spread between listeners, and each one bounds a claim made somewhere else on this site.

A trill is half a consonance unless it borrows its own resolution

A struck suspension loses roughness because its tied note is old, and the obvious repairs are all things a player already has. Three of them fail in ways nobody would predict. A re-strike under about nine decibels below the chord is smoother than the tie it replaces. No rate of trill stops the fading, because the three notes under the trill are what is fading. And a trill on the upper neighbour spends half its time on a chord note: only the trill that alternates a suspension with its own resolution gives back more than a single strike.

One walk round the bearing measures the piano and the tuner together

A tuner who counts twelve fourths and fifths to the printed table and comes home flat has measured something. Divide the closing beat's rate by three and correct the whole table by that share, and the worst interval improves from a hundred per cent off to about twenty, on any piano. It keeps the table with one number. And the number cannot tell a stiff string from a slow count: a tuner counting nine per cent slow gets the right bearing from the wrong table, and one correction removes that luck along with the error.

A vibraphone's valve is a door on the sound and a drain on the bar

A vibraphone's motor is usually described as a duty cycle: loud while the valve is open and quiet while it is shut, with a tremolo as deep as the gap between them. Put the tube's own time back and the description comes apart. The door and the drain obey different laws. A low note's tremolo is a third shallower than a high note's at the same speed, the note rings almost three times longer than with the valve left open and well beyond any mean of the two states, and the same stroke can land eight decibels apart according to where the motor happens to be.